Learn Mathematics

Jordan Theorem for Dirichlet Integrals Examples 1

Recall from the Jordan's Theorem for Dirichlet Integrals page that if we consider the Dirichlet integral, $\displaystyle{\int_0^b g(t) \frac{\sin \alpha t}{t} \: dt}$ then if $g$ is of bounded variation on $[0, b]$ for some $b > 0$ then we saw that:

Note that $g$ is of bounded variation for all $b > 0$ since $g$ is a polynomial and every polynomial is of bounded variation on any closed and bounded interval $[0, b]$. So, by Jordan's test for Dirichlet integrals we have that: